Golomb-Dickman Constant Continued Fraction

The Golomb-Dickman constant continued fraction is the simple continued fraction of the Golomb-Dickman constant , which begins (OEIS A225336). In the first 14510 terms, 41.6% are 1, 16.8% are 2, and so on (E. Weisstein, Jul. 25, 2013).

Positions are counted from 0 at the initial term . The plot above shows the positions of the first occurrences of 1, 2, 3, ... in the continued fraction, the first few of which are 1, 8, 9, 30, 25, 18, 110, 242, 59, 100, 12, 71, 28, 153, 225, 114, 159, 66, ... (OEIS A225364). The smallest positive integers not appearing in the first 14510 terms of the continued fraction are 90, 108, 110, 124, ... (E. W. Weisstein, Jul. 25, 2013).

The successive running maxima in the continued fraction are 0, 1, 22, 28, 43, 48, 66, 491, 1706, 4763, 38371, ... (OEIS A225337), which occur at positions 0, 1, 6, 24, 39, 50, 52, 72, 259, 1002, 4610, ... (OEIS A225363).

Let the continued fraction of be denoted and let the denominators of the convergents be denoted , , ..., . Then plots above show successive values of , , , which appear to converge to Khinchin's constant (left figure) and , which appear to converge to the Lévy constant (right figure), although neither of these limits has been rigorously established.